Optimal. Leaf size=304 \[ \frac{b (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right ) (A b (b c (1-m)-a d (5-m))+a B (a d (3-m)+b c (m+1)))}{2 a^2 e (m+1) (b c-a d)^3}-\frac{d (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right ) (a d (A d (1-m)+B c (m+1))+b c (B c (3-m)-A d (5-m)))}{2 c^2 e (m+1) (b c-a d)^3}+\frac{d (e x)^{m+1} (a A d-2 a B c+A b c)}{2 a c e \left (c+d x^2\right ) (b c-a d)^2}+\frac{(e x)^{m+1} (A b-a B)}{2 a e \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)} \]
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Rubi [A] time = 0.802401, antiderivative size = 304, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {579, 584, 364} \[ \frac{b (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right ) (A b (b c (1-m)-a d (5-m))+a B (a d (3-m)+b c (m+1)))}{2 a^2 e (m+1) (b c-a d)^3}-\frac{d (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right ) (a d (A d (1-m)+B c (m+1))+b c (B c (3-m)-A d (5-m)))}{2 c^2 e (m+1) (b c-a d)^3}+\frac{d (e x)^{m+1} (a A d-2 a B c+A b c)}{2 a c e \left (c+d x^2\right ) (b c-a d)^2}+\frac{(e x)^{m+1} (A b-a B)}{2 a e \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 579
Rule 584
Rule 364
Rubi steps
\begin{align*} \int \frac{(e x)^m \left (A+B x^2\right )}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx &=\frac{(A b-a B) (e x)^{1+m}}{2 a (b c-a d) e \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac{\int \frac{(e x)^m \left (2 a A d-A b c (1-m)-a B c (1+m)-(A b-a B) d (3-m) x^2\right )}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx}{2 a (b c-a d)}\\ &=\frac{d (A b c-2 a B c+a A d) (e x)^{1+m}}{2 a c (b c-a d)^2 e \left (c+d x^2\right )}+\frac{(A b-a B) (e x)^{1+m}}{2 a (b c-a d) e \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac{\int \frac{(e x)^m \left (2 \left (A \left (4 a b c d-b^2 c^2 (1-m)-a^2 d^2 (1-m)\right )-a B c (b c+a d) (1+m)\right )-2 b d (A b c-2 a B c+a A d) (1-m) x^2\right )}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{4 a c (b c-a d)^2}\\ &=\frac{d (A b c-2 a B c+a A d) (e x)^{1+m}}{2 a c (b c-a d)^2 e \left (c+d x^2\right )}+\frac{(A b-a B) (e x)^{1+m}}{2 a (b c-a d) e \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac{\int \left (\frac{2 b c (-A b (b c (1-m)-a d (5-m))-a B (a d (3-m)+b c (1+m))) (e x)^m}{(b c-a d) \left (a+b x^2\right )}+\frac{2 a d (b c (B c (3-m)-A d (5-m))+a d (A d (1-m)+B c (1+m))) (e x)^m}{(b c-a d) \left (c+d x^2\right )}\right ) \, dx}{4 a c (b c-a d)^2}\\ &=\frac{d (A b c-2 a B c+a A d) (e x)^{1+m}}{2 a c (b c-a d)^2 e \left (c+d x^2\right )}+\frac{(A b-a B) (e x)^{1+m}}{2 a (b c-a d) e \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac{(b (A b (b c (1-m)-a d (5-m))+a B (a d (3-m)+b c (1+m)))) \int \frac{(e x)^m}{a+b x^2} \, dx}{2 a (b c-a d)^3}-\frac{(d (b c (B c (3-m)-A d (5-m))+a d (A d (1-m)+B c (1+m)))) \int \frac{(e x)^m}{c+d x^2} \, dx}{2 c (b c-a d)^3}\\ &=\frac{d (A b c-2 a B c+a A d) (e x)^{1+m}}{2 a c (b c-a d)^2 e \left (c+d x^2\right )}+\frac{(A b-a B) (e x)^{1+m}}{2 a (b c-a d) e \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac{b (A b (b c (1-m)-a d (5-m))+a B (a d (3-m)+b c (1+m))) (e x)^{1+m} \, _2F_1\left (1,\frac{1+m}{2};\frac{3+m}{2};-\frac{b x^2}{a}\right )}{2 a^2 (b c-a d)^3 e (1+m)}-\frac{d (b c (B c (3-m)-A d (5-m))+a d (A d (1-m)+B c (1+m))) (e x)^{1+m} \, _2F_1\left (1,\frac{1+m}{2};\frac{3+m}{2};-\frac{d x^2}{c}\right )}{2 c^2 (b c-a d)^3 e (1+m)}\\ \end{align*}
Mathematica [A] time = 0.271041, size = 207, normalized size = 0.68 \[ \frac{x (e x)^m \left (-(b c-a d) \left (a^2 d (A d-B c) \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )+b c^2 (A b-a B) \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )\right )+a^2 c d \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right ) (a B d-2 A b d+b B c)-a b c^2 \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right ) (a B d-2 A b d+b B c)\right )}{a^2 c^2 (m+1) (a d-b c)^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.056, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( B{x}^{2}+A \right ) \left ( ex \right ) ^{m}}{ \left ( b{x}^{2}+a \right ) ^{2} \left ( d{x}^{2}+c \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B x^{2} + A\right )} \left (e x\right )^{m}}{{\left (b x^{2} + a\right )}^{2}{\left (d x^{2} + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (B x^{2} + A\right )} \left (e x\right )^{m}}{b^{2} d^{2} x^{8} + 2 \,{\left (b^{2} c d + a b d^{2}\right )} x^{6} +{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} + 2 \,{\left (a b c^{2} + a^{2} c d\right )} x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B x^{2} + A\right )} \left (e x\right )^{m}}{{\left (b x^{2} + a\right )}^{2}{\left (d x^{2} + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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